Scaling of Superradiant Peak Emission in Spatially Extended Emitter Arrays

Abstract

In quantum optics, superradiance is a phenomenon in which a system of $N$ fully excited quantum emitters radiate intense flashes of light during collective decay. However, computing its peak intensity exactly for many spatially separated emitters remains challenging due to the exponential growth of the underlying Hilbert space with system size $N$. Based on third-order cumulant expansion methods, we present general scaling laws for the expononent of the peak emission rate as a function of the emitter number in free-space emitter arrays and arrays coupled to one-dimensional waveguide reservoirs. We find, that for 1D chains in free-space the peak emission rate scales linearly with $N$, while for 2D and 3D arrays with finite emitter spacing it scales superlinearly but sub-quadratically. For emitter chains coupled to waveguide reservoirs we find that the peak emission rate scales quadratically with $N$.

Figures (4)

• Figure 1: (a) Schematic of a spatially extended ensemble of $N$ two-level quantum emitters with spontaneous decay rate $\Gamma$ and finite nearest-neighbor separation $a > 0$. When the fully excited state $|e\rangle^{\otimes N}$ undergoes superradiant decay, the photon emission rate $R$ (in photons per second) reaches a pronounced peak $R_\mathrm{peak}$ ($\alpha \!> \!1$), departing from monotonic or exponential decay ($\alpha \!= \!1$). (b) In this work we study spatially extended arrays in free-space or coupled to bidirectional waveguide reservoirs based on a third-order cumulant expansion method.
• Figure 2: Peak photon emission rate as a function of the emitter number $N$. (a) one-dimensional euqidistant chain, (b) two-dimensional square array (c) three-dimensional cubic array. For the chain we find a linear scaling of the peak emission rate with $N$ while for the square and cubic arrays the peak emission rate scales superlinearly but less than quadratically at finite emitter separations. (d) Equidistant emitter chain coupled to a bidirectional waveguide with relative phases between the emitters, $ka/\pi=0.1,0.2,0.3$. We find, that $R_\mathrm{peak}\propto \Gamma N^2$, irrespective of the relative phase. In (a) for nearest-neighbor spacings $a/\lambda_0=0.1,0.15,0.2$, in (b-c) $a/\lambda_0=0.15,0.2,0.3$. Circles ($\circ$) correspond to circular $(1,\pm i,0)/\sqrt{2}$ and squares ($\square$) to linear polarized $(0,0,1)$ dipoles. For all plots a third-order cumulant expansion has been used.
• Figure 3: The exponent $\alpha$, assuming the general form for the peak emission rate $R_\mathrm{peak} = \Gamma \beta N^\alpha$ with the same emitter spacings as in Fig. \ref{['fig2']}. One-dimensional chain for (a) circular and (b) linear polarized emitters with the exponent $\alpha$ converging to one for increasing $N$. Square array for (c) circular and (d) linear polarization. Cubic array for (e) circular and (f) linear polarization. The continuous lines (-) correspond to a third-order cumulant expansion and dashed lines (- -) to a second-order cumulant expansion.
• Figure 4: The exponent $\alpha$ as a function of emitter spacing $a$ for (a) linear chain, (b) square array and (c) cubic array for linear and circular polarization. The simulations are performed with a third order cumulant expansion. We note, that the exponent will keep decreasing for for increasing emitter numbers at any given spacing.